Almost-Fisher families

A classic theorem in combinatorial design theory is Fisher's inequality, which states that a family $\mathcal F$ of subsets of $[n]$ with all pairwise intersections of size $\lambda$ can have at most $n$ non-empty sets. One may weaken the condition by requiring that for every set in $\mathcal F$, all but at most $k$ of its pairwise intersections have size $\lambda$. We call such families $k$-almost $\lambda$-Fisher. Vu was the first to study the maximum size of such families, proving that for $k=1$ the largest family has $2n-2$ sets, and characterising when equality is attained. We substantially refine his result, showing how the size of the maximum family depends on $\lambda$. In particular we prove that for small $\lambda$ one essentially recovers Fisher's bound. We also solve the next open case of $k=2$ and obtain the first non-trivial upper bound for general $k$.Comment: 27 pages (incluiding one appendix

An Energy and Performance Exploration of Network-on-Chip Architectures

In this paper, we explore the designs of a circuit-switched router, a wormhole router, a quality-of-service (QoS) supporting virtual channel router and a speculative virtual channel router and accurately evaluate the energy-performance tradeoffs they offer. Power results from the designs placed and routed in a 90-nm CMOS process show that all the architectures dissipate significant idle state power. The additional energy required to route a packet through the router is then shown to be dominated by the data path. This leads to the key result that, if this trend continues, the use of more elaborate control can be justified and will not be immediately limited by the energy budget. A performance analysis also shows that dynamic resource allocation leads to the lowest network latencies, while static allocation may be used to meet QoS goals. Combining the power and performance figures then allows an energy-latency product to be calculated to judge the efficiency of each of the networks. The speculative virtual channel router was shown to have a very similar efficiency to the wormhole router, while providing a better performance, supporting its use for general purpose designs. Finally, area metrics are also presented to allow a comparison of implementation costs

Magic State Distillation with Low Space Overhead and Optimal Asymptotic Input Count

We present an infinite family of protocols to distill magic states for $T$-gates that has a low space overhead and uses an asymptotic number of input magic states to achieve a given target error that is conjectured to be optimal. The space overhead, defined as the ratio between the physical qubits to the number of output magic states, is asymptotically constant, while both the number of input magic states used per output state and the $T$-gate depth of the circuit scale linearly in the logarithm of the target error $\delta$ (up to $\log \log 1/\delta$). Unlike other distillation protocols, this protocol achieves this performance without concatenation and the input magic states are injected at various steps in the circuit rather than all at the start of the circuit. The protocol can be modified to distill magic states for other gates at the third level of the Clifford hierarchy, with the same asymptotic performance. The protocol relies on the construction of weakly self-dual CSS codes with many logical qubits and large distance, allowing us to implement control-SWAPs on multiple qubits. We call this code the "inner code". The control-SWAPs are then used to measure properties of the magic state and detect errors, using another code that we call the "outer code". Alternatively, we use weakly-self dual CSS codes which implement controlled Hadamards for the inner code, reducing circuit depth. We present several specific small examples of this protocol.Comment: 39 pages, (v2) renamed "odd" and "even" weakly self-dual CSS codes of (v1) to "normal" and "hyperbolic" codes, respectively. (v3) published in Quantu

A matrix interpolation between classical and free max operations: I. The univariate case

Recently, Ben Arous and Voiculescu considered taking the maximum of two free random variables and brought to light a deep analogy with the operation of taking the maximum of two independent random variables. We present here a new insight on this analogy: its concrete realization based on random matrices giving an interpolation between classical and free settings.Comment: 14 page

Experimental Works and Analysis to Determine the Degradation of Permanent Magnet Force under Cyclic Magnetic Field

Permanent magnet is widely used in today’s application such motors we used in car or generator in power stations. Materials commonly used are neodymium, alnico, and also soft ferrites. Its unique ability to retain its magnet force for such a long time is one of the attributes it excels in. However, loss of magnet’s strength or demagnetization can bring major impact in applications. External field or electric current is one of the method where a magnet can be magnetized and demagnetized by altering the number of domains of electrons which the mechanism of magnetism is present. Permanent magnet – permanent magnet interactions in cyclic motion resembles what happen in both motor and generators. Their magnetic strength is indeed all acting differently before, during and after the application to be specific, in cyclic motion. Hence, further research must be done followed by in-depth analysis to know how these magnets would react under cyclic motion

A Novel Seed Based Random Interleaving for OFDM System and Its PHY Layer Security Implications

Wireless channels are characterized by multipath and fading that can often cause long burst of errors. Even though, to date, many very sophisticated error correcting codes have been designed, yet none can handle long burst of errors efficiently. An interleaver, a device that distributes a burst of errors, possibly caused by a deep fade, and makes them appear as simple random errors, therefore, proves to a very useful technique when used in conjunction with an efficient error correcting code. In this work, a novel near optimal seed based random interleaver is designed. An optimal interleaver scatters a given burst of errors uniformly over a fixed block of data - a property that is measured by so called ‘spread’. The design makes use of a unique seed based pseudo-random sequence generator or logistic map based chaotic sequence generator to scramble the given block of data. Since the proposed design is based on a seed based scrambler, the nature of input is irrelevant. Therefore, the proposed interleaver can interleave either the bits or the symbols or the packets or even the frames. Accordingly, in this work, we analyze the suitability of interleaver when introduced before or after the modulation in single carrier communication systems and show that interleaving the bits before modulation or interleaving the symbols after modulation has same advantage. We further show that, in an orthogonal frequency division multiplexing (OFDM) systems, the position of interleaver, whether before or after constellation mapper, has no significance, and is interchangeable. However, scrambling symbols is computationally less expensive than scrambling bits. For the purpose of analyzing the performance of the proposed seed based random interleaver, simulations are carried out in MATLAB®. Results show that our proposed seed based random interleaver has near optimal properties of ‘spread’ and ‘dispersion’. Furthermore, the proposed interleaver is evaluated in terms of bit error rate (BER) versus length of burst error in a single carrier system both before and after modulation. The proposed interleaver out-performs the built in RANDINTLV in MATLAB® when used in the same system. It shows that proposed interleaver can convert greater amount of burst errors into simple random errors than that of MATLAB® interleaver. The proposed interleaver is also tested in IEEE 802.16e based WiMAX system with Stanford University Interim (SUI) channels to compare the performance of average BER versus SNR for both pre modulation and post modulation interleaver. Results show that pre modulation interleaver and post modulation has same performance. There is also a side advantage of this seed based interleaver, in that it generates a variety of unique random-looking interleaving sequences. Only a receiver that has the knowledge of the input seed can generate this sequence and no one else. If the interleaving patterns are kept secure then it can possibly be used to introduce an extra layer of security at physical (PHY) layer. In that way, at PHY layer, one builds an additional entry barrier to break through and it comes with no extra cost. This property has been investigated by carrying out key sensitivity analysis to show that the attacks to guess key can be very futile, as difference at 4th decimal place in the initial condition can lead to entirely different scrambling